Archive for Mathematical Logic

, Volume 56, Issue 3–4, pp 187–213 | Cite as

Chains of saturated models in AECs

  • Will Boney
  • Sebastien VaseyEmail author


We study when a union of saturated models is saturated in the framework of tame abstract elementary classes (AECs) with amalgamation. We prove:

Theorem 0.1. If K is a tame AEC with amalgamation satisfying a natural definition of superstability (which follows from categoricity in a high-enough cardinal), then for all high-enough \(\lambda {:}\)
  1. (1)

    The union of an increasing chain of \(\lambda \)-saturated models is \(\lambda \)-saturated.

  2. (2)

    There exists a type-full good \(\lambda \) -frame with underlying class the saturated models of size \(\lambda \).

  3. (3)

    There exists a unique limit model of size \(\lambda \).

Our proofs use independence calculus and a generalization of averages to this non first-order context.


Abstract elementary classes Forking Independence calculus Classification theory Stability Superstability Tameness Saturated models Limit models Averages Stability theory inside a model 

Mathematics Subject Classification

Primary 03C48 Secondary 03C47 03C52 03C55 03E55 


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  1. 1.
    Albert, M.H., Grossberg, R.: Rich models. J. Symb. Log. 55(3), 1292–1298 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Baldwin, J.T.: Categoricity, University Lecture Series, vol. 50, American Mathematical Society (2009)Google Scholar
  3. 3.
    Boney, W., Grossberg, R.: Forking in short and tame AECs. Ann. Pure Appl. Log. (2017). doi: 10.1016/j.apal.2017.02.002
  4. 4.
    Boney, W., Grossberg, R., Kolesnikov, A., Vasey, S.: Canonical forking in AECs. Ann. Pure Appl. Log. 167(7), 590–613 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Boney, W.: Tameness and extending frames. J. Math. Log. 14(2), 1450007 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Boney, W.: Tameness from large cardinal axioms. J. Symb. Log. 79(4), 1092–1119 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Drueck, F.: Limit models, superlimit models, and two cardinal problems in abstract elementary classes, Ph.D. thesis (2013).
  8. 8.
    Grossberg, R., Lessmann, O.: Shelah’s stability spectrum and homogeneity spectrum in finite diagrams. Arch. Math. Log. 41(1), 1–31 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Grossberg, R.: A Course in Model Theory I (in preparation)Google Scholar
  10. 10.
    Grossberg, R.: On chains of relatively saturated submodels of a model without the order property. J. Symb. Log. 56, 124–128 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grossberg, R.: Classification theory for abstract elementary classes. Contemp. Math. 302, 165–204 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Grossberg, R., VanDieren, M.: Galois-stability for tame abstract elementary classes. J. Math. Log. 6(1), 25–49 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Grossberg, R., VanDieren, M., Villaveces, A.: Uniqueness of limit models in classes with amalgamation. Math. Log. Q. 62, 367–382 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Harnik, V.: On the existence of saturated models of stable theories. Proc. Am. Math. Soc. 52, 361–367 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Makkai, M., Shelah, S.: Categoricity of theories in \({L}_{\kappa,\omega }\), with \(\kappa \) a compact cardinal. Ann. Pure Appl. Log. 47, 41–97 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Shelah, S.: The lazy model theoretician’s guide to stability. Log. Anal. 18, 241–308 (1975)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Shelah, S.: Classification of non elementary classes II. Abstract elementary classes. In: Baldwin, J.T. (ed.) Classification Theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, pp. 419–497. Springer, Berlin (1987)Google Scholar
  18. 18.
    Shelah, S.: Classification theory and the number of non-isomorphic models. In: Studies in Logic and the Foundations of Mathematics, vol. 92, 2nd edn. North-Holland, Amsterdam (1990)Google Scholar
  19. 19.
    Shelah, S.: Categoricity for abstract classes with amalgamation. Ann. Pure Appl. Log. 98(1), 261–294 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Shelah, S.: Classification Theory for Abstract Elementary Elasses, Studies in Logic: Mathematical Logic and Foundations, vol. 18. College Publications, Norcross (2009)Google Scholar
  21. 21.
    Shelah, S.: Classification Theory for Abstract Elementary Classes 2, Studies in Logic: Mathematical Logic and Foundations, vol. 20. College Publications, Norcross (2009)Google Scholar
  22. 22.
    Shelah, S., Villaveces, A.: Toward categoricity for classes with no maximal models. Ann. Pure Appl. Log. 97, 1–25 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    VanDieren, M.: Categoricity in abstract elementary classes with no maximal models. Ann. Pure Appl. Log. 141, 108–147 (2006)MathSciNetCrossRefGoogle Scholar
  24. 24.
    VanDieren, M.: Erratum to “Categoricity in abstract elementary classes with no maximal models” [Ann. Pure Appl. Logic 141 (2006) 108–147]. Ann. Pure Appl. Log. 164(2), 131–133 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Vasey, S.: Building independence relations in abstract elementary classes. Ann. Pure Appl. Log. 167(11), 1029–1092 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Vasey, S.: Forking and superstability in tame AECs. J. Symb. Log. 81(1), 357–383 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Vasey, S.: Infinitary stability theory. Arch. Math. Log. 55, 567–592 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsHarvard UniversityCambridgeUSA
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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